What is the general solution for (cos(x))/(sin(2x))= 5/7 ?

The general solution for (cos(x))/(sin(2x))= 5/7 is x=0.77539…+2pin,x=pi-0.77539…+2pin

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(cos(x))/(sin(2x))= 5/7

Solution

\frac{cos ( x )}{sin ( 2 x )} = \frac{5}{7}

Solution

x = 0.77539 \dots + 2 πn, x = π - 0.77539 \dots + 2 πn

Degrees

x = 44.42700 \dots^{\circ} + 36 0^{\circ} n, x = 135.57299 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{cos ( x )}{sin ( 2 x )} = \frac{5}{7}

Subtract

\frac{5}{7}

from both sides

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7} = 0

Simplify

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7} : \frac{7 cos ( x ) - 5 sin ( 2 x )}{7 sin ( 2 x )}

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7}

Least Common Multiplier of

sin (2 x), 7 : 7 sin (2 x)

sin (2 x), 7

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (2 x)

7

= 7 sin (2 x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

7 sin (2 x)

For

\frac{cos ( x )}{sin ( 2 x )} :

multiply the denominator and numerator by

7

\frac{cos ( x )}{sin ( 2 x )} = \frac{cos ( x ) \cdot 7}{sin ( 2 x ) \cdot 7}

For

\frac{5}{7} :

multiply the denominator and numerator by

sin (2 x)

\frac{5}{7} = \frac{5 sin ( 2 x )}{7 sin ( 2 x )}

= \frac{cos ( x ) \cdot 7}{sin ( 2 x ) \cdot 7} - \frac{5 sin ( 2 x )}{7 sin ( 2 x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ( x ) \cdot 7 - 5 sin ( 2 x )}{7 sin ( 2 x )}

\frac{7 cos ( x ) - 5 sin ( 2 x )}{7 sin ( 2 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

7 cos (x) - 5 sin (2 x) = 0

Rewrite using trig identities

- 5 sin (2 x) + 7 cos (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 5 \cdot 2 sin (x) cos (x) + 7 cos (x)

Simplify

= - 10 sin (x) cos (x) + 7 cos (x)

7 cos (x) - 10 cos (x) sin (x) = 0

Factor

7 cos (x) - 10 cos (x) sin (x) : - cos (x) (10 sin (x) - 7)

7 cos (x) - 10 cos (x) sin (x)

Factor out common term

- cos (x)

= - cos (x) (- 7 + 10 sin (x))

- cos (x) (10 sin (x) - 7) = 0

Solving each part separately

cos (x) = 0 or 10 sin (x) - 7 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

10 sin (x) - 7 = 0 : x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

10 sin (x) - 7 = 0

Move

7

to the right side

10 sin (x) - 7 = 0

Add

7

to both sides

10 sin (x) - 7 + 7 = 0 + 7

Simplify

10 sin (x) = 7

10 sin (x) = 7

Divide both sides by

10

10 sin (x) = 7

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{7}{10}

Simplify

sin (x) = \frac{7}{10}

sin (x) = \frac{7}{10}

Apply trig inverse properties

sin (x) = \frac{7}{10}

General solutions for

sin (x) = \frac{7}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

Show solutions in decimal form

x = 0.77539 \dots + 2 πn, x = π - 0.77539 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (cos(x))/(sin(2x))= 5/7 ?
The general solution for (cos(x))/(sin(2x))= 5/7 is x=0.77539…+2pin,x=pi-0.77539…+2pin